Example of a semisimple Lie algebra with degenerate Killing form

Question

We know that when the killing form of a Lie algebra is nondegenerate then it is semisimple. I am looking for a semisimple Lie algebra with degenerate killing form. I know if the field is of characteristic zero it is impossible to find one.

Answer 1

In the exercises of Humphreys book on Lie algebras and Representation theory the example of $\mathfrak{sl}(3)$ (respectively of $\mathfrak{sl}(3)/Z(\mathfrak{sl}(3))$ in characteristic $3$ is worked out. Its Killing form is identical zero, but $\mathfrak{sl}(3)/Z(\mathfrak{sl}(3))$ is still a simple Lie algebra in characteristic $3$. One can find all details in the solutions here. The matrix of the Killing form relative to the standard basis is given by $$ \begin{pmatrix} 12 & -6 & 0 & 0 & 0 & 0 & 0 & 0\cr -6 & 12 & 0 & 0 & 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 \cr 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 \cr 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 \cr 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 \end{pmatrix} $$ which is identical zero for $3=0$.